Problem: In the diagram, if $\angle PQR = 48^\circ$, what is the measure of $\angle PMN$? [asy]

size(6cm);

pair p = (0, 0); pair m = dir(180 - 24); pair n = dir(180 + 24); pair r = 1.3 * dir(24); pair q = 2 * 1.3 * Cos(48) * dir(-24);

label("$M$", m, N); label("$R$", r, N); label("$P$", p, 1.5 * S); label("$N$", n, S); label("$Q$", q, SE);

draw(m--q--r--n--cycle);

add(pathticks(m--p, s=4));

add(pathticks(n--p, s=4));

add(pathticks(r--p, 2, spacing=0.9, s=4));

add(pathticks(r--q, 2, spacing=0.9, s=4));

[/asy]
Solution: In $\triangle PQR$, since $PR=RQ$, then $\angle RPQ=\angle PQR = 48^\circ$.

Since $\angle MPN$ and $\angle RPQ$ are opposite angles, we have $\angle MPN = \angle RPQ=48^\circ$.

In $\triangle PMN$, $PM=PN$, so $\angle PMN = \angle PNM$.

Therefore, $$\angle PMN = \frac{1}{2}(180^\circ - \angle MPN) = \frac{1}{2}(180^\circ - 48^\circ) = \frac{1}{2}(132^\circ)=\boxed{66^\circ}.$$